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Boundedness of certain oscillatory singular integrals. (English) Zbl 0886.42008
The authors prove the $$L^p(\mathbb{R}^n)$$ ($$1<p<\infty$$) and $$H^1(\mathbb{R}^n)$$ boundedness of oscillatory singular integral operators $$T$$. These operators are defined by $Tf(x)= \text{p.v.} \int_{\mathbb{R}^n}e^{i\Phi(x-y)}K(x-y)f(y) dy$ where $$K(x)$$ is a Calderón-Zygmund kernel and the phase function $$\Phi\in C^\infty(\mathbb{R}^n\setminus\{0\})$$ is a real-valued function satisfying the conditions $|D^\alpha\Phi(x)|\leq C|x|^{a-|\alpha|},\quad |\alpha|\leq3, \qquad \sum_{|\alpha|=2}|D^\alpha\Phi(x)|\geq C'|x|^{a-2},$ where $$a\in\mathbb{R}$$ is some fixed number and the constants $$C, C'>0$$ are independent of $$x\in\mathbb{R}^n\setminus\{0\}$$ (as usual $$\alpha=(\alpha_1,\dots,\alpha_n)\in(\mathbb{N}_0)^n$$, $$|\alpha|=\alpha_1+\cdots+\alpha_n$$, $$D^\alpha=(\frac\partial{\partial x_1})^{\alpha_1}\cdots (\frac\partial{\partial x_n})^{\alpha_n}$$).
The results of the papers imply that for the typical example of the phase function $$\Phi(x)=|x|^a$$, the $$L^p(\mathbb{R}^n)$$ boundedness of $$T$$ holds when $$a\neq 0$$ or $$a\neq1=n$$, while the $$H^1(\mathbb{R}^n)$$ boundedness of $$T$$ takes place when $$a\neq 0,1$$. Recall that for $$a=0$$ the operator $$T$$ is the usual Calderón-Zygmund singular integral operator whose behaviour on the spaces $$H^1(\mathbb{R}^n)$$ and $$L^p(\mathbb{R}^n)$$ is well known. For $$a=n=1$$ $$T$$ is known to be unbounded on both $$L^p(\mathbb{R}^n)$$ and $$H^1(\mathbb{R}^n)$$, for $$a=1$$, $$n\geq1$$ the $$H^1(\mathbb{R}^n)$$ boundedness of $$T$$ does not hold [cf. P. Sjölin, J. Lond. Math. Soc., II. Ser. 23, 442-454 (1981; Zbl 0449.46051)].
A similar problem in the context of the Besov space $$B_0^{1,0}(\mathbb{R}^n)$$ was studied earlier by one of the authors [D. Fan, “An oscillating integral in the Besov $$B^{1,0}_0(\mathbb{R}^n)$$” (to appear)].
Reviewer: P.Gurka (Praha)

##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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